Asymptotic Behavior for Nonlinear Systems of Phase Transitions
$+\ovalbox{\tt\small REJECT}\star B^{B_{1\backslash }k_{\backslash }}\mathfrak{F}^{j\backslash }\dagger F$ $ffi\ovalbox{\tt\small REJECT}$ $\ovalbox{\tt\small REJECT} ffl$ (Naoki Sato)
$\mp\overline{\not\in}XB\mathfrak{B}_{\backslash \backslash }E\neq$ $E|7J\langle$ $\Gamma$ (Jun Shirohzu)
$\mp\ovalbox{\tt\small REJECT} X\ovalbox{\tt\small REJECT}\Leftrightarrow$ $\ovalbox{\tt\small REJECT} E$ $P_{\overline{R}}\not\equiv$ ( Nobuyuki Kenmochi)
1. Introduction
We consider the following nonlinear system:
$\frac{\partial\rho(u)}{\partial t}+\frac{\partial w}{\partial t}-\triangle u-=f(t,x)$ in $Q$ $:=(0, +\infty, )\cross\Omega$, (1.1)
$\nu\frac{\partial w}{\partial t}+\beta(w)+g(w)\ni u$ in $Q$ (1.2)
with lateral boundary condition:
$\frac{\partial u}{\partial n}+\alpha_{N}(x)u=h_{N}(t, x)$
and initial conditions:
on $\Sigma$ $:=(0, +\infty)\cross\Gamma$, (1.3)
$u(0, \cdot)=u_{0}$, $w(0, \cdot)=w_{0}$ in $\Omega$. (1.4)
Here $\Omega$ is a bounded domain in $R^{N}(N\geq 1)$ with smooth boundary $\Gamma$
$:=\partial\Omega;\rho$ is a
monotone increasing and bi-Lipschitz continuous function on $R;\nu$ is a positive constant;
$\beta$ is a maximal monotone graph in $R\cross R;g$ is a smooth function defined on $R;\alpha_{N}$ is a
non-negative, bounded and measurable function on $\Gamma$ such that
$\alpha_{N}>0$ on a subset of $\Gamma$
with positive measure; $f,$$h_{N},$$u_{0}$ and $w_{0}$ are given data.
For simplicity problem $(1.1)-(1.4)$ is denoted by (CP). This is a simplified model for
a class of solid-liquid phase change problems, and in this context $u$ represents a function
related to temperature and $w$ a non-conserved order parameter (the state variable
charac-terizing phase). For instance, we have the following examples:
(1) Stefan problemwith phase relaxation, in which $\beta$is the subdifferential of the indicator
function of the interval $[0,1]$ and $g\equiv 0$
.
This case was discussed as a melting problemwith supercooling and superheating effect in [12,5].
(2) Phase-field model with constraint, in which $\beta$ is the same as in (1), $\rho(u)=u,$ $g(w)=$
side of (1.2). This is a phase-field model with constraint $0\leq w\leq 1$ and was discussed
in [6,9,11]. We may consider system $(1.1)-(1.4)$ as an approximation of this problem
with small $\kappa>0$.
Furthermore we refer to [2,1] for papers dealing with similar problems.
In this paper, we discuss the large-time behavior of the solution $\{u, w\}$. In fact, under the condition that $f(t, x)arrow f^{\infty}(x)$ and $h_{N}(t, x)arrow h_{N}^{\infty}(x)$ in an appropriate sense as $tarrow+\infty$, it will be shown that as $tarrow+\infty,$ $u(t, \cdot)$ and $w(t, \cdot)$ converge to a solution
$\{u^{\infty}, w^{\infty}\}$ of the corresponding steady-state problem
$\{\begin{array}{l}-\triangle u^{\infty}=f^{\infty}(x) in \Omega,\frac{\partial u^{\infty}}{\partial n}+\alpha_{N}(x)u^{\infty}=h_{N}^{\infty}(x)on \Gamma,\beta(w^{\infty})+g(w^{\infty})\ni u^{\infty} in \Omega.\end{array}$
2. Existence and uniqueness result for (CP)
Problem (CP) is discussed under the following assumptions (Al)$-(A6)$:
(Al) $\rho$ : $Rarrow R$ is an increasing and bi-Lipschitz continuous function.
(A2) $\beta$ is a maximal monotone graph in $R\cross R$ such that for some numbers $\sigma_{*},$$\sigma^{*}$ with
$-\infty<\sigma_{*}<\sigma^{*}<+\infty$
$\overline{D(\beta)}=[\sigma_{*}, \sigma^{*}]$;
note in this case that $R(\beta)=R$, so that there is a non-negative proper l.s.$c$. convex
function $\hat{\beta}$ on
$R$ whose subdifferential $\partial\hat{\beta}$ coincides with
$\beta$ in $R$, and in the context
of solid-liquid system we can consider that $w=\sigma_{*}$ (resp. $\sigma^{*}$) indicates the pure solid (resp. liquid) phase and any intermediate value $w$ indicates a state ofmixture. (A3) $g$ : $Rarrow R$ is a Lipschitz continuous function with compact support in $R$; in this
case note that there is a non-negative primitive $\hat{g}$ of $g$.
(A4) $f\in L_{l\circ c}^{2}(R_{+};L^{2}(\Omega))$.
(A5) $h_{N}\in W_{loc}^{1,2}(R_{+};L^{2}(\Gamma))$ with $\sup_{t\geq 0}|h_{N}|_{W^{1,2}(t,t+1;L^{2}(\Gamma))}<+\infty$. (A6) $u_{0}\in L^{2}(\Omega)$ and $w_{0}\in L^{2}(\Omega)$ with $\hat{\beta}(w_{0})\in L^{1}(\Omega)$.
We introduce some function spaces and a convex function in order to discuss (CP) in
the framework of abstract evolution equations of the form $U’(t)+\partial\varphi^{t}(U(t))+G(U(t))\ni\tilde{f}(t)$.
Let $V:=H^{1}(\Omega)$ with norm
$|z|_{V}:= \{|\nabla z|_{L^{2}(\Omega)}^{2}+\int_{\Gamma}\alpha_{N}|z|^{2}d\Gamma\}^{\frac{1}{2}}$,
and denote by $V^{*}$ the dual space of $V$ and by $\langle\cdot,$ $\cdot)$ the duality pairing between $V^{*}$ and $V$.
Then, identifying $L^{2}(\Omega)$ with its dual space by means of the usual inner product
$(v, z)$ $:= \int_{\Omega}$vzdx,
we see that
$V\subset L^{2}(\Omega)\subset V^{*}$
with compact injections.
Let $F$ be the duality mapping from $V$ onto $V^{*}$ which is given by the formula
$\langle Fv,$$z \rangle=\int_{\Omega}\nabla v\cdot\nabla zdx+\int_{\Gamma}\alpha_{N}vzd\Gamma$ for any $v,$$z\in V$
.
It is easy to see that $V^{*}$ becomes a Hilbert space with inner product $(\cdot,$$\cdot)_{*}$ given by
$(v, z)_{*}:=\langle v,$$F^{-1}z\rangle(=\langle z, F^{-1}v\rangle)$ for any $v,$$z\in V^{*}$.
Now, consider the product space
$X:=V^{*}\cross L^{2}(\Omega)$,
which becomes a Hilbert space with inner product $(\cdot,$ $\cdot)_{X}$ given by
$([e_{1}, w_{1}], [e_{2}, w_{2}])_{X}$ $:=\cdot(e_{1}, e_{2})_{*}+\nu(w_{1},w_{2})$ for any $[e_{i}, w_{i}]\in X(i=1,2)$
.
Next, given the boundary data $h_{N}$, choose $h:R+arrow H^{1}(\Omega)$ such that for each $t\geq 0$
$\int_{\Omega}\nabla h(t)\cdot\nabla zdx+\int_{\Gamma}\alpha_{N}h(t)zd\Gamma=\int_{\Gamma}h_{N}(t)zd\Gamma$ for all $z\in V$;
note from (A5) that $\sup_{t\geq 0}|h|_{W^{1,2}(t,t+1;H^{1}\langle\Omega))}<+\infty$
.
Also, using $h$ and $\hat{\beta}$, for each
$t\geq 0$, define a proper l.s.$c$. convexfunction $\varphi^{t}$ on $X$ by
$\varphi^{t}(U)=\{\begin{array}{l}\int_{\Omega}\rho^{*}(e-w)dx+\int_{\Omega}\hat{\beta}(w)dx-(h(t), e)if U=[e, w]\in L^{2}(\Omega)\cross L^{2}(\Omega) with \hat{\beta}(w)\in L^{1}(\Omega),+\infty otherwise,\end{array}$
where $\rho^{*}$ is a non-negative primitive of $\rho^{-1}$
.
We denote by$\partial\varphi^{t}$ the subdifferential of $\varphi^{t}$ in
$X$ and its characterization is given by the following theorem.
Theorem 2.1. $($
cf.
$[5_{f}9])$ Let $t\geq 0,$ $[e^{*}, w^{*}]\in X$ and $[e, w]\in D(\partial\varphi^{t})$. Then $[e^{*}, w^{*}]\in$$(a)$ $e^{*}=F(\rho^{-1}(e-w)-h(t))$, that is, $\rho^{-1}(e-w)-h(t)\in V$ and
$\langle e^{*},$$z \rangle=\int_{\Omega}\nabla(\rho^{-1}(e-w)-h(t))\cdot\nabla zdx+\int_{\Gamma}\alpha_{N}(\rho^{-1}(e-w)-h(t))zd\Gamma$
for
all $z\in V$;$(b)$ there exists a
function
$\xi\in L^{2}(\Omega)$ such that $\xi\in\beta(w)a.e$. on $\Omega$ and$\nu w^{*}=\xi-\rho^{-1}(e-w)$ in $L^{2}(\Omega)$.
Moreover,
for
$U_{i}^{*}=[e_{i}^{*}, w_{i}^{*}]\in\partial\varphi^{t}(U_{i})$ with $U_{i}=[e_{i}, w_{i}]\in D(\partial\varphi^{t})(i=1,2)$, $(U_{1}^{*}-U_{2}^{*}, U_{1}-U_{2})_{X}=|(e_{1}-w_{1})-(e_{2}-w_{2})|_{L^{2}(\Omega)}+(\xi_{1}-\xi_{2}, w_{1}-w_{2})$, where $\xi_{i}\in L^{2}(\Omega)$ is as anyfunction
$\xi$ in $(b)$for
each $i=1,2$.A weak formulation for (CP) is given as follows.
Definition 2.1. A couple $\{u, w\}$ of functions $u$ : $R+arrow V^{*}$ and $w$ : $R+arrow L^{2}(\Omega)$ is
called a (weak) solution of (CP) on $R_{+}$, if the following conditions (wl)$-(w3)$ are fulfilled
for any finite $T>0$:
(wl) $\rho(u)\in C([0, T];V^{*})\cap W_{loc}^{1,2}((0, T];V^{*})\cap L^{2}(0, T;L^{2}(\Omega)),$ $u\in L_{loc}^{2}((0, T];H^{1}(\Omega))$, $w\in C([0, T];L^{2}(\Omega))\cap W_{loc}^{1,2}((0, T];L^{2}(\Omega))$, and $\hat{\beta}(w)\in L^{1}(0, T;L^{1}(\Omega))$.
(w2) $\rho(u)(0)=p(u_{0})$ and
$\langle u’(t)+w’(t),$ $z \rangle+\int_{\Omega}\nabla(u(t)-h(t))\cdot\nabla zdx+\int_{\Gamma}\alpha_{N}(u(t)-h(t))zd\Gamma=(f(t), z)$ for all $z\in V$ and a.e. $t\in[0, T]$, where the prime ’ denotes the derivative in time.
(w3) there exists $\xi\in L_{loc}^{2}((0, T];L^{2}(\Omega))$ such that $\xi\in\beta(w)$ a.e. on QT $:=(0, T)\cross\Omega$ and
$\nu(w’(t), z)+(\xi(t)+g(w(t)), z)=(u(t), z)$
for all $z\in L^{2}(\Omega)$ and a.e. $t\in[0, T]$.
According to Theorem 2.1, (CP) can be reformulated as an evolution equation in $X$ in the following form:
$\{\begin{array}{ll}U’(t)+\partial\varphi^{t}(U(t))+G(U(t))\ni\tilde{f}(t), in X, t\geq 0,U(0)=[\rho(u_{0})+w_{0}, w_{0}], \end{array}$
As to the solvability of (CP) we have:
Theorem 2.2. (cf. [6,9]) Assume that $(A 1)-(A6)$ hold. Then,
for
any $T>0,$ $(CP)$ admitsone and only one solution $\{u, w\}$ on $[0, T]$ such that
$\{\begin{array}{ll}t^{\frac{1}{2}}\rho(u)’\in L^{2}(0, T;V^{*}), t^{\frac{1}{2}}u\in L^{2}(0, T;H^{1}(\Omega)),t\rho(u)’\in L^{2}(0, T;L^{2}(\Omega)), tu\in L^{\infty}(0, T;H^{1}(\Omega)),\end{array}$
$t^{\frac{1}{2}}w’\in L^{2}(0, T;L^{2}(\Omega))$, $t\hat{\beta}(w)\in L^{\infty}(0, T;L^{1}(\Omega))$,
$t^{\frac{1}{2}}\xi\in L^{2}(0, T;L^{2}(\Omega))$
where $\xi$ is the
function
in condition $(w3)$.3. Large-time behavior of the solution
Further suppose that there are $h_{N}^{\infty}\in L^{2}(\Gamma)$ and $f^{\infty}\in L^{2}(\Omega)$ such that
$h_{N}-h_{N}^{\infty}\in L^{2}(R_{+};L^{2}(\Gamma))$, $f-f^{\infty}\in L^{2}(R_{+};L^{2}(\Omega))$, (3.1)
and consider the steady-state problem $(3.2)-(3.3)$:
$-\triangle u^{\infty}=f^{\infty}(x)$ in $\Omega$, $\frac{\partial u^{\infty}}{\partial n}+\alpha_{N}(x)u^{\infty}=h_{N}^{\infty}(x)$ on $\Gamma$, (3.2) $\beta(w^{\infty})+g(w^{\infty})\ni u^{\infty}$ in $\Omega$. (3.3)
We should note that problem (3.2) does not include $w^{\infty}$, and it has a unique solution
$u^{\infty}\in H^{1}(\Omega)$ in the variational sense, i.e.,
$\int_{\Omega}\nabla(u^{\infty}-h^{\infty})\cdot\nabla zdx+\int_{\Gamma}\alpha_{N}(u^{\infty}-h^{\infty})zd\Gamma=(f^{\infty}, z)$ for all $z\in V$, (3.4) where $h^{\infty}\in H^{1}(\Omega)$ such that
$\int_{\Omega}\nabla h^{\infty}\cdot\nabla zdx+\int_{\Gamma}\alpha_{N}h^{\infty}zd\Gamma=\int_{\Gamma}h_{N}^{\infty}zd\Gamma$ for all $z\in V$.
We see from (3.1) that $h-h^{\infty}\in L^{2}(R_{+};H^{1}(\Omega))$.
In the sequelwemean by $(P^{\infty})$ thealgebraic relation (3.3) with the solution $u^{\infty}\in H^{1}(\Omega)$
of (3.4), and $w^{\infty}=w^{\infty}(x)$ is called a solution of $(P^{\infty})$.
As the following example shows, the steady-state problem $(P^{\infty})$ has in general infinitely
many solutions.
Example 3.1. Consider the case when
$f^{\infty}(x)\equiv 0$, $h_{N}^{\infty}(x)\equiv l_{0}$, $\alpha_{N}(x)\equiv 1$, $\beta=\partial I_{[-1,1]}$ and $g(w)=w^{3}-w$
(i) when $l_{0}> \frac{2}{3\sqrt{3}}$ $($resp. $l_{0}<- \frac{2}{3\sqrt{3}})$, the algebraic relation
$\beta(r)+g(r)\ni u^{\infty}(=l_{0})$ (3.5) has exactly one solution $r=1$ (resp. $-1$).
(ii) when $l_{0}= \frac{2}{3\sqrt{3}}$ (resp. $- \frac{2}{3\sqrt{3}}$), $(3.5)$ has exactly two solutions $r=- \frac{1}{\sqrt{3}}$ (resp. $\frac{1}{\sqrt{3}}$), $1$
(resp. $-1$).
(iii) when $|l_{0}|<-372_{3},$ $(3.5)$ has exactly three solutions $r=\xi_{-},\xi_{0},$$\xi_{+}$ with $-1\leq\xi_{-}<\xi_{0}<$
$\xi+\leq 1$.
Physically (i) means that if the temperature is kept high (resp. low) enough, then the limit state $(as tarrow+\infty)$ will be of pure liquid (resp. solid). On the other hand, (ii)
and (iii) mean that if the temperature is kept near the phase transition temperature, then the limit state possibly includes a mushy region. In particular, in the case of (iii), all step functions $w^{\infty}$ with rangein $\{\xi_{-}, \xi_{0}, \xi_{+}\}$ are solutions of$(P^{\infty})$ and hence $(P^{\infty})$ has in general
infinitely many solutions.
Our main result is stated in the following theorem.
Theorem 3.1. Suppose that conditions $(A1)-(A\theta)$ and (3.1) hold, and let $\{u, w\}$ be the
solution to $(CP)$ on $R_{+}$. Further, suppose that
for
each $p\in R$ the (algebraic) inclusion$\beta(r)+g(r)\ni p$
has a
finite
numberof
solutions $r$ in $\overline{D(\beta)}$. Then,$u(t)arrow u^{\infty}$ weakly in $H^{1}(\Omega)$ as $tarrow+\infty$, (3.6)
where $u^{\infty}$ is the unique solution
of
(3.4), and there exists afunction
$w^{\infty}\in L^{\infty}(\Omega)$ such that$\beta(w^{\infty}(x))+g(w^{\infty}(x))\ni u^{\infty}(x)$
for
$a.e$. $x\in\Omega$and
$w(t, x)arrow w^{\infty}(x)$
for
$a.e$. $x\in\Omega$ as $tarrow+\infty$.We prove the theorem by the following four lemmas.
Lemma 3.1. Under the same assumptions
of
Theorem 3.1,for
the solution $\{u, w\}$ to $(CP)$on $R+$’ we have
$u-u_{\infty}\in L^{2}(R_{+};H^{1}(\Omega)),$ $w’\in L^{2}(R_{+};L^{2}(\Omega))$ and $\hat{\beta}(w)\in L^{\infty}(R_{+};L^{1}(\Omega))$, (3.7)
Proof. Multiplying the difference of (1.1) and (3.2) by $u(t)-u^{\infty}$ and (1.2) by $w’(t)$, we
get
$\frac{d}{dt}\{\int_{\Omega}\rho^{*}(\rho(u(t)))dx-(\rho(u(t))+w(t), u^{\infty})\}+(w’(t), u(t))+$
$+| \nabla(u(t)-u^{\infty})|_{L^{2}\langle\Omega)}^{2}+\int_{\Gamma}\alpha_{N}|u(t)-u^{\infty}|^{2}d\Gamma$
$=(f(t)-f^{\infty}, u(t)-u^{\infty})+ \int_{\Gamma}(h(t)-h^{\infty})(u(t)-u^{\infty})d\Gamma$ and
$\nu|w’(t)|_{L^{2}(\Omega)}^{2}+\frac{d}{dt}\{\int_{\Omega}\hat{\beta}(w(t))dx+\int_{\Omega}\hat{g}(w(t))dx\}=(u(t), w’(t))$
for a.e. $t\geq 0$. Adding these two equalities we have
$\frac{d}{dt}\{\int_{\Omega}\rho^{*}(\rho(u(t)))dx-(\rho(u(t))+w(t), u^{\infty})+\int_{\Omega}\hat{\beta}(w(t))dx+\int_{\Omega}\hat{g}(w(t))dx\}+$
$+ \nu|w’(t)|_{L^{2}(\Omega)}^{2}+|\nabla(u(t)-u^{\infty})|_{L^{2}(\Omega)}^{2}+\int_{\Gamma}\alpha_{N}|u(t)-u^{\infty}|^{2}d\Gamma$
$=(f(t)-f^{\infty}, u(t)-u^{\infty})+ \int_{\Gamma}(h(t)-h^{\infty})(u(t)-u^{\infty})d\Gamma$ for a.e. $t\geq 0$, so that there are positive constants $C_{1}$ and $C_{2}$ such that
$\frac{d}{dt}\{\int_{\Omega}\rho^{*}(\rho(u(t)))dx-(\rho(u(t))+w(t), u^{\infty})+\int_{\Omega}\hat{\beta}(w(t))dx+\int_{\Omega}\hat{g}(w(t))dx\}+$
$+\nu|w’(t)|_{L^{2}(\Omega)}^{2}+C_{1}|u(t)-u^{\infty}|_{H^{1}(\Omega)}^{2}$
$\leq C_{2}\{|f(t)-f^{\infty}|_{L^{2}(\Omega)}^{2}+|h(t)-h^{\infty}|_{L^{2}(\Gamma)}^{2}\}$
for a.e. $t\geq 0$. Therefore, for all $T>0$, we have
$\int_{\Omega}\rho^{*}(\rho(u(T)))dx-(\rho(u(T))+w(T), u^{\infty})+\int_{\Omega}\hat{\beta}(w(T))dx+\int_{\Omega}\hat{g}(w(T))dx+$
$+ \nu\int_{0}^{T}|w’(t)|_{L^{2}(\Omega)}^{2}dt+C_{1}\int_{0}^{T}|u(t)-u^{\infty}|_{H^{1}(\Omega)}^{2}dt$
$\leq C_{2}\{\int_{0}^{T}|f(t)-f^{\infty}|_{L^{2}(\Omega)}^{2}dt+\int_{0}^{T}|h(t)-h^{\infty}|_{L^{2}(\Gamma)}^{2}dt\}+$
$+ \int_{\Omega}\rho^{*}(\rho(u_{0}))dx-(\rho(u_{0})+w_{0}, u^{\infty})+\int_{\Omega}\hat{\beta}(w_{0})dx+\int_{\Omega}\hat{g}(w_{0}))dx$.
Hence(3.7) is obtained. Also, (3.8) is adirect consequence of(3.7) andastandard regularity
result for parabolic equations. $\square$
Lemma 3.2. Under the same assumptions
of
Theorem 3.1, putThen, $U^{t}(x)arrow 0$ as $tarrow+\infty$
for
$a.e$. $x\in\Omega$.Proof. By Lemma 3.1, we have
$\lim_{T\nearrow+\infty}\int_{T}^{+\infty}dt\int_{\Omega}|w’(t, x)|^{2}dx=0$,
so that
$\lim_{T\nearrow+\infty}\int_{\Omega}dx\int_{T}^{+\infty}|w’(t, x)|^{2}dt=0$.
Hence,
$\int_{T}^{+\infty}|w’(t, x)|^{2}dtarrow 0$ as $Tarrow+\infty$ for a.e. $x\in\Omega$.
This implies the lemma. $\square$
Lemma 3.3. Under the same assumptions
of
Theorem $3.1_{f}(3.6)$ holds.Proof. Let $\{u, w\}$ be a solution to (CP) and $u^{\infty}$ be the solution to (3.4).
Let $\{t_{n}\}$ be any sequence with $t_{n}arrow+\varphi$ as $narrow+\infty$, and put
$u_{n}(t)$ $:=u(t_{n}+t),$ $w_{n}(t)$ $:=w(t_{n}+t),$ $f_{n}(t)$ $:=f(t_{n}+t),$ $h_{n}(t)$ $:=h(t_{n}+t)$
for $0\leq t\leq 1$.
Since by Lemma 3.1, $u-u^{\infty}$ and $w’$ are in $L^{2}(R_{+};H^{1}(\Omega))$ and $L^{2}(R_{+};L^{2}(\Omega))$, respectively,
we see that
$u_{n}arrow u^{\infty}$ in $L^{2}(0,1;H^{1}(\Omega))$, (3.9) and
$w_{n}’arrow 0$ in $L^{2}(0,1;L^{2}(\Omega))$, (3.10)
as $narrow+\infty$. Moreover since by Lemma 3.1, $u$ is bounded in $H^{1}(\Omega)$ on $[1,$$+\infty)$, we may
assume that for a function $\tilde{u}^{\infty}$ in $H^{1}(\Omega)$
$\acute{u}_{n}(0)=u(t_{n})arrow\tilde{u}^{\infty}$ weakly in $H^{1}(\Omega)$ (3.11)
as $narrow+\infty$. Now, consider the Cauchy problem for each $n$
$\{\begin{array}{ll}\rho(u_{n})’(t)+\partial\Phi_{n}^{t}(u_{n}(t))=f_{n}(t)-w_{n}’(t) in L^{2}(\Omega), 0\leq t\leq 1,u_{n}(0)=u(t_{n}) \end{array}$
where $\Phi_{n}^{t}$ is a proper l.s.$c$. and convex function on $L^{2}(\Omega)$ suchthat for each $n$ and $t\in[0,1]$
$\Phi_{n}^{t}(z):=\{\begin{array}{ll}\frac{1}{2}|z-h_{n}(t)|_{V}^{2} if z\in V,+\infty otherwise,\end{array}$
and $\partial\Phi_{n}^{t}$ is the subdifferential of $\Phi_{n}^{t}$ in $L^{2}(\Omega)$. From (3.9), (3.10) and (3.11), we see that
$f_{n}-w_{n}’arrow f^{\infty}$ in $L^{2}(0,1;L^{2}(\Omega))$
and
$u_{n}(0)arrow\tilde{u}^{\infty}$ in $L^{2}(\Omega)$
as $narrow+\infty$, where $\Phi_{\infty}$ is a proper l.s.$c$. and convex function on $L^{2}(\Omega)$ such that
$\Phi_{\infty}(z):=\{\begin{array}{ll}\frac{1}{2}|z-h^{\infty}|_{V}^{2} if z\in V,+\infty otherwise.\end{array}$
Therefore, by a general theory in [8],
$u_{n}arrow\tilde{u}$ in $C([0,1];L^{2}(\Omega))$ as $tarrow+\infty$, (3.12)
where $\tilde{u}$ is the solution of
$\{\begin{array}{ll}\rho(\tilde{u})’(t)+\partial\Phi_{\infty}(\tilde{u}(t))=f^{\infty} in L^{2}(\Omega), 0\leq t\leq 1,\tilde{u}(0)=\tilde{u}^{\infty^{-}}.\end{array}$
From (3.9) and (3.12) it follows that $\tilde{u}=u^{\infty}$ on $[0,1]$. Consequently (3.6) holds. $\square$
Lemma 3.4. Under the same assumptions
of
Theorem $3.1_{f}$ put$V(x)$ $:=$
{
$r\in\overline{D(\beta)};w(t_{n},$$x)arrow r$for
some $t_{n}$ with $t_{n}arrow+\infty$}
for
$x\in\Omega$.Then,
(1) $V(x)\neq\emptyset$
for
$a.e$. $x\in\Omega$;(2) $\beta(r)+g(r)\ni u^{\infty}(x)$
for
all $r\in V(x)$ and $a.e$. $x\in\Omega$;(3) $V(x)$ is a singleton
for
$a.e$. $x\in\Omega$.Proof. (1) is clear by the boundedness of$w(t, x)$ on $R$.
Let $x\in\Omega$ with $\lim_{tarrow+\infty}l^{t+1}|w’(\tau, x)|^{2}d\tau=0$ (cf. Lemma 3.2) and $r\in V(x)$. Then, there
exists a sequence $\{t_{n}\}$ such that
$t_{n}arrow+\infty$ and $w(t_{n}, x)arrow r$ as $narrow+\infty$.
Fixing $x$, put
$w_{n}(t):=w(t_{n}+t, x),$ $u_{n}(t):=u(t_{n}+t, x)$ for $0\leq t\leq 1$.
By Lemma 3.3 and (A2), we may assume that
Now consider a sequence of ODEs:
$\{$
$w_{n}’(t)+\beta(w_{n}(t))+g(w_{n}(t))\ni u_{n}(t)$ for $0\leq t\leq 1$,
By a general theory in [8] again, $w_{n}$ converges in $C([0,1])$ to the solution $\tilde{w}$ of
$\{\begin{array}{ll}w_{n}(0)=w(t_{n}, x).\tilde{w}’(t)+\beta(\tilde{w}(t))+g(\tilde{w}(t))\ni u^{\infty}(x) for 0\leq t\leq 1,\tilde{w}(0)=r.\end{array}$
(3.13)
But, since $\tilde{w}’\equiv 0$ i.e. $\tilde{w}\equiv r$ by assumption, we see from (3.13) that
$\beta(r)+g(r)\ni u^{\infty}(x)$.
Thus, (2) is proved. At last, we show (3). Suppose that $V(x)$ has more than two elements
for some $x\in\Omega$, say $r_{1},$$r_{2}\in V(x),$ $r_{1}<r_{2}$
.
By definition, there exists two sequence $\{s_{n}\}$and $\{t_{n}\}$ such that
$s_{n}arrow+\infty,$ $w(s_{n}, x)arrow r_{1}$, $t_{n}arrow+\infty,$ $w(t_{n}, x)arrow r_{2}$ as $narrow+\infty$, and
$s_{n}<t_{n}<s_{n+1}<t_{n+1}$ for $n=1,2,3,$$\cdots$ .
From the continuity of$w$ with respect to $t$, for any$r\in(r_{1}, r_{2})$, there exists a sequence $\{\tau_{n}\}$
with $\tau_{n}arrow+\infty(narrow+\infty)$ such that
$s_{n}<\tau_{n}<t_{n}$ for $n=1,2,3,$$\cdots$ and $w(\tau_{n}, x)=r$ for large $n$.
This implies that $r\in V(x)$ and hence $[r_{1}, r_{2}]\subset V(x)$. This contradicts the assumption
that $\beta(r)+g(r)\ni u^{\infty}(x)$ has a finite number of solutions $r$ in $\overline{D(\beta)}$. Thus, $V(x)$ must be
a singleton for a.e. $x\in\Omega$. $\square$
Inparticular, (2) and (3) of Lemma3.4imply that $w(t, x)$ convergesto a solution $w^{\infty}(x)$
for a.e. $x\in\Omega$ as $tarrow+\infty$ and the limit $w^{\infty}$ is a solution of $(P^{\infty})$. Thus we complete the
proof of Theorem 3.1.
References
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Naoki SATO, Jun SHIROHZU
Department of Mathematics
Graduate School of Science and Technology
Chiba University
263 Chiba, Japan Nobuyuki KENMOCHI
Department of Mathematics, Faculty ofEducation
Chiba University
